If the transfer case is fine, check the tranny fluid, and for leaks under the vehicle.
You may be able to leave the vehicle running for a bit and it will work. Then it def a fluid issue. The fluid is pumped through the radiator to keep it cool, but it also warms it up after the vehicle reaches temp. An important detail for me up here when the temp is below zero. At 200k I have to baby my tranny a bit.